The Jeju death case seems like it might be a suicide

To determine if it was homicide, one would have to look for existing grudges, such as romantic entanglements or debt, etc.
Since none of these have surfaced, the individual was suffering from depression, and the rope used was reportedly brought from home, it doesn't seem like a random local crime.
If there were a suspect, it would be a different story, but without one and if no further evidence emerges, the possibility of suicide seems high, and frankly, there isn't much way to prove otherwise.
Many people have been playing detective, asking, "Who on earth commits suicide in a forest?" but they say that after one's residence, forests and mountains are the most common locations for suicide.
In the UK, it is said to be around 5–10%, though there are no relevant statistics for Korea.
It is said that there are actually quite a few people who go to Jeju for that purpose.